Berry Curvature and Four-Dimensional Monopoles in the Relativistic Chiral Kinetic Equation

Abstract

We derive a relativistic chiral kinetic equation with manifest Lorentz covariance from Wigner functions of spin-1/2 massless fermions in a constant background electromagnetic field. It contains vorticity terms and a 4-dimensional Euclidean Berry monopole which gives axial anomaly. By integrating out the zero-th component of the 4-momentum p, we reproduce the previous 3-dimensional results derived from the Hamiltonian approach, together with the newly derived vorticity terms. The phase space continuity equation has an anomalous source term proportional to the product of electric and magnetic fields (FσρF. This provides a unified interpretation of the chiral magnetic and vortical effects, chiral anomaly, Berry curvature, and the Berry monopole in the framework of Wigner functions. Introduction. -The Berry phase is a topological phase factor acquired by an eigen-energy state when it undergoes adiabatic evolution along a loop in parameter space [1]. It is in close analogy to the Aharonov-Bohm phase when a charged particle moves in a loop enclosing a magnetic flux, while the Berry curvature is like the magnetic field. The integral of the Berry curvature over a closed surface can be quantized as integers known as Chern-Simons numbers, which is similar to the Dirac magnetic monopole and has deep connection with the quantum Hall effect. The Berry phase is a beautiful, simple and universal structure in quantum physics and has many interesting applications, for a recent review of the Berry phase in condensed matter physics, see e.g. Ref.[2].

Chiral drag force

“…These anomalous contributions to the hydrodynamic motion of a chiral fluid have been discussed widely [9,12,. These effects originate in topological aspects of the gauge theory [12,14,15,33,36,37], as is of course the case for the axial anomaly itself [44].…”

Chiral vortical and magnetic effects in the anomalous transport model

“…More realistic studies of CMW and CVW in heavy ion collisions have been carried out using the anomalous hydrodynamics, which extends the normal hy-drodynamics by including also the axial charge current in QGP [9]. Also, chiral kinetic equations, which include the anomalous transport of massless fermions and the non-equilibrium effect, have been developed [14][15][16][17][18][19]. However, different methods have been proposed in including the chiral vortical effect in the chiral kinetic approach.…”

From Chirality of Neutrinos to Supernovae

“…For more details of the chiral kinetic theory, see Refs. [3][4][5][6]. In supernovae, there are not only neutrinos, but also ordinary matter (nuclei and electrons), which must also be incorporated in the transport theory.…”

“…In the language of kinetic theory for neutrinos, these topological effects can be captured by the notion of the Berry curvature. The kinetic framework with Berry curvature corrections for chiral fermions (not limited to neutrinos) is now referred to as the chiral kinetic theory [3][4][5][6].…”